Transcript Wake field

Collective effects
Erik Adli, University of Oslo, August 2015, [email protected], v2.12
Introduction
•
Particle accelerators are continuously
being pushed to new parameter regimes
with higher currents, higher power and
higher intensity
•
Performance is usually limited by multiparticle effects, including collective
instabilities and collisions
•
A core element of particle accelerator
physics is the study of collective effects.
The understanding of collective
instabilities has made it possible to
overcome limitations and increase
performance significantly
•
We here describe the two most common
effects, space charge and wake fields, in
some detail.
Example of increase in beam intensity in the
CERN Proton Synchrotron the last decades
Intense beams: example
Compressed high energy electron beam (at the “FACET” facility at SLAC) :
The beam can be defined by the following parameters :
• Charge per bunch: Q = 3 nC (N = 2 x 1010 electrons)
• Emittance: enx,y = 100 um
• Beam energy: E = 20 GeV (v ≈ c)
Focused size at interaction point :
• sx = 20 um, sy = 20 um, sz = 20 um (Gaussian)
Other key beam characteristics deduced from the above accelerator parameters:
• Peak current: Ipeak = Qpeak/t = Qc / √2p sz = 19 kA (cf. IAlfven = 17 kA)
• Beam density: ne ≈ N / 4/3p(sx sy sz) = 6 x 1017 / cm3
Plasma description:
• Electron transverse energy: Ee = ½ mvx2 = ½ m c2 x’2 ~ 1 meV
• Electron (transverse) temperature: Te ~ me / kB c2 x’2 ~ 100 K
• Debye length (shielding length): lD = √(e0kBT/e2n0) ~ 1 nm
-> lD << sx, sy, sz (collective effects dominate over collisional effects)
Space charge
A general definition of “Space Charge” : a collection of particles with a net
electric charge occupying a region, either in free space or in a device.
In accelerators, we usually use the term space charge to describe the
collective field/force in a same-charge particle bunch.
Consider the bunch from the previous example, at rest. We may do a
rough estimate of the space charge field using a two-particle model:
~2sx,y
This is a very large field (cf. accelerating fields discussed earlier). Using the
two particle model, we may estimate the force, and the time scale for the
bunch to blow-up due to space charge forces to sub-picosecond. How can
we accelerator dense particles bunches at all?
( Later, we give a more detailed calculations, assuming a Gaussian
distribution. )
F
+ Q/2
+ Q/2
F
Coordinates – speed of light frame
x
z = s - vt
s
The beam travels in the s direction, with speed v. The co-moving coordinate z = s-vt is
defined in a frame following the beam travelling with speed v , and gives thus the relative
position inside the beam. In plasma wakefield applications, the beam is often travelling with
velocity v = c. In this case the frame where z = s- ct is defined is called the speed of light
frame.
In accelerator physics we usually describe fields and charges in the laboratory frame,
however, following the co-moving frame. NB: this is not the same as describing the fields
and charges in the rest frame of the beam, as quantities described in the co-moving frame
are not Lorenz transformed with respect to the lab frame.
Space charge – moving charges
Two-particle interaction :
xlab
x’
x
v2
v1
z
y
z
s
y'
Fields transforms to the lab frame as :
We observe beams, fields and forces in the lab frame. The force on particle 1 is
F = e(E+v1 x B). Particle 2 generates no magnetic field in its rest frame, which gives
the relation (BT – v2/c2 x E) = 0. The total transverse force on particle 1 in the lab
frame is thus F = e(E - v1 x (v2/c2 x E)), or for parallel velocities :
Fx,y = eEx,y (1±v1v2/c2)
For v1 = v2 = v we calculate the relativistic space charge suppression :
Fx,y = eEx,y (1-v2/c2) = eEx,y / g2
EM fields from a relativistic particle
Fields in particle’s rest frame
Fields in lab frame
Lorentz Tranformations
v=0
Ultra-relativistic limit (v=c) :
Spherical symmetrical field
Field compression
-> “Pancake field”
Space Charge – using Gauss’ law
Alternate calculation: Gauss law's. Gauss law is valid
also for relativistic moving (or accelerating) charges. In
the rest frame, the beam sees an electrostatic field. In the
lab frame, the moving charges produce a magnetic field.
We assume a uniform density beam in shape of a
cylinder, with beam charge density r = Ne / pa2L :
r
Gauss law gives :
E 
r
r
Ampere’s law gives :
Bf 
F
+
+
F
Rest frame
FE
FB
+
FB
+
FE
Lab frame
2e 0
r v
r
2
2e 0 c
Combine terms Fr = e(Er - vBf) :
+
+
This is the direct space charge effect. “Direct” : does not take into
account the effects of conducting walls surrounding the beam.
c
c
Lab frame, v = c
v
Space Charge for Gaussian Beams
KSC ~ Ipeak / (bg)3
“beam
perveance”
• Typical lattice focusing (FODO with ~10 m between magnets): <b> ~ 10 m -> Kb = 1/100 m-2
• For our example 20 GeV electron beam (a few slides ago) : KSC = 5 x 10-5 m-2 << Kb
-> space charge completely suppressed at high Lorenz factor
• For the same electron beam, but at 20 MeV : KSC = 5 x 104 m-2 >> Kb
Summary
K. Schindl
Linear defocusing. Gives tune shift
in rings. Can be compensated by
stronger lattice focusing.
Non-linear defocusing. Gives tune
shift and tune spread in rings.
Beam-beam effects
Relativistic space charge suppression holds only for equal charge moving at the
same velocity, in the same direction (even then, only holds fully in free space).
These requirement are often violated. An important example is the Beam-Beam
interaction :
For v1 and v2 of opposite sign in our previous calculation (5 slides ago) we get :
Fx,y = eEx,y (1+v1v2/c2)
•
•
Two colliding beams see the field of each other before collision. They may
be strongly attracted, and may deform.
Important limitation for collider luminosity. Considered the main challenge
for LHC luminosity. We will revisit the topic of beam beam effects in the
linear collider lectures.
Wake fields
Most of the material is G. Rumolo’s slides from
CAS Course on wake fields
Wake fields (general)
Source, q1
Witness, q2
2b
z
z
z
L
– While source and witness ( qi d(s-ct) ), at a distance z, move centered in a
perfectly conducting chamber, the witness does not feel any force (g >> 1)
– When the source encounters a discontinuity (e.g., transition, device), it
produces an electromagnetic field, which trails behind (wake field)
o The source loses energy
o The witness feels a net force all along an effective length of the structure, L
13
Wake field characteristics
Wake field framework approximations
When calculating wake fields in accelerator, the calculations are greatly simplified,
allowing for simple descriptions of complex fields, by the following two approximations :
1) Rigid beam approximation: bunch static over length wake field is calculated
2) Impulse approximation: we only care about integrated force over the length wake
field is calculated, not the details of the fields in time and space
We will define wake function (wake potential) as such integrated quantities.
Wake fields (general)
Source, q1
Witness, q2
2b
z
L
– Not only geometric discontinuities cause electromagnetic fields trailing
behind sources traveling at light speed.
– For example, a pipe with finite conductivity causes a delay in the induced
currents, which also produces delayed electromagnetic fields
o No ringing, only slow decay
o The witness feels a net force all along an effective length of the structure, L
– In general, also electromagnetic boundary conditions can be the origin of
wake fields.
16
dp/p0
z
1. The longitudinal plane
Longitudinal wake function: definition
Source, q1
Witness, q2
z
2b
L
18
Longitudinal wake function: properties
– The value of the wake function in 0, W||(0), is related to the energy lost
by the source particle in the creation of the wake
– W||(0)>0 since DE1<0
– W||(z) is discontinuous in z=0 and it vanishes for all z>0 because of the
ultra-relativistic approximation
W||(z)
z
19
The energy balance
What happens to the energy lost by the source?
– In the global energy balance, the energy lost by the source splits into
o Electromagnetic energy of the modes that remain trapped in the object
⇒ Partly dissipated on
or into purposely designed inserts or
⇒ Partly transferred to following particles (or the same particle over
successive turns), possibly feeding into an instability
o Electromagnetic energy of modes that propagate down the beam chamber
(above cut-off), which will be eventually lost on surrounding lossy materials
20
The energy balance
What happens to the energy lost by the source?
– In the global energy balance, the energy lost by the source splits into
o Electromagnetic energy of the modes that remain trapped in the object
⇒ Partly dissipated on
or into purposely designed inserts or
⇒ PartlyThe
transferred
to loss
following
particles
(or the same particle over
energy
is very
important
successive turns), possibly feeding into an instability!
because
o Electromagnetic energy of modes that propagate down the beam chamber
⇒which
It causes
beam induced
heating lossy materials
(above cut-off),
will be eventually
lost on surrounding
of the beam environment
(damage, outgassing)
⇒ It feeds into both longitudinal
and transverse instabilities
through the associated EM fields
21
Longitudinal impedance
– The wake function of an accelerator component is basically its Green function in
time domain (i.e., its response to a pulse excitation)
⇒ Very useful for macroparticle models and simulations, because it can be
used to describe the driving terms in the single particle equations of
motion!
– We can also describe it as a transfer function in frequency domain
– This is the definition of longitudinal beam coupling impedance of the element
under study
[W]
[W/s]
22
Longitudinal impedance: resonator
W||
Re[Z||]
Im[Z||]
wr
T=2p/wr
– The frequency wr is related to the oscillation of Ez, and therefore to the frequency
of the mode excited in the object
– The decay time depends on how quickly the stored energy is dissipated
(quantified by a quality factor Q)
23
Longitudinal impedance: cavity
– A more complex example: a simple pill-box
cavity with walls having finite conductivity
– Several modes can be excited
– Below the pipe cut-off frequency the
width of the peaks is only determined
by the finite conductivity of the walls
– Above, losses also come from
propagation in the chamber
Re[Z||]
Im[Z||]
24
Single bunch effects
W||
Im[Z||]
Re[Z||]
25
Single bunch effects
W||
26
Multi bunch effects
Re[Z||]
Im[Z||]
27
Multi bunch effects
W||
28
Multi bunch effects
Dz
29
Example: the Robinson instability
– To illustrate the Robinson instability we will use some simplifications:
⇒ The bunch is point-like and feels an external linear force (i.e. it
would execute linear synchrotron oscillations in absence of the
wake forces)
⇒ The bunch additionally feels the effect of a multi-turn wake
dp/p0
z
Unperturbed: the bunch executes
synchrotron oscillations at ws
30
The Robinson instability
– To illustrate the Robinson instability we will use some simplifications:
⇒ The bunch is point-like and feels an external linear force (i.e. it
would execute linear synchrotron oscillations in absence of the
wake forces)
⇒ The bunch additionally feels the effect of a multi-turn wake
dp/p0
z
The perturbation also changes
the oscillation amplitude
Unstable motion
The perturbation changes ws
31
The Robinson instability
– To illustrate the Robinson instability we will use some simplifications:
⇒ The bunch is point-like and feels an external linear force (i.e. it
would execute linear synchrotron oscillations in absence of the
wake forces)
⇒ The bunch additionally feels the effect of a multi-turn wake
dp/p0
z
The perturbation also changes
the oscillation amplitude
Damped motion
32
2. The transverse plane
Transverse wake function: definition
Source, q1
Witness, q2
z
2b
L
– In an axisymmetric structure (or simply with a top-bottom and left-right symmetry) a
source particle traveling on axis cannot induce net transverse forces on a witness
particle also following on axis
– At the zero-th order, there is no transverse effect
– We need to introduce a breaking of the symmetry to drive transverse effect, but at
the first order there are two possibilities, i.e. offset the source or the witness
34
Transverse dipolar wake function:
definition
Source, q1
Witness, q2
Dx1 (or Dy1)
z
2b
L
35
Transverse dipolar wake function
– The value of the transverse dipolar wake functions in 0, Wx,y(0), vanishes because
source and witness particles are traveling parallel and they can only – mutually –
interact through space charge, which is not included in this framework
– Wx,y(0--)<0 since trailing particles are deflected toward the source particle (Dx1 and
Dx’2 have the same sign)
– Wx,y(z) has a discontinuous derivative in z=0 and it vanishes for all z>0 because of the
ultra-relativistic approximation
Wx,y(z)
z
36
Discrete approximation of cavity
transverse impedance
Impedances in frequency domain
calculated using electromagnetic
solvers. Impedance can be
approximated using a number of
discrete modes (above 9 modes used).
From CLIC decelerator design
Dipole wake instabilities in linacs
Single bunch: head drives tail resonantly ->
banana shape, beam-break up
Two-bunches: one
bunch drives the
second resonantely
Linac Beam breakup growth factor:
~NWs/kb
N: charge
W: dipole wake
s: distance
kb: betatron k
Rings: A glance into the head-tail modes
• Different transverse head-tail modes correspond to different parts of the
bunch oscillating with relative phase differences. E.g.
– Mode 0 is a rigid bunch mode
– Mode 1 has head and tail oscillating in counter-phase
– Mode 2 has head and tail oscillating in phase and the bunch center in
opposition
39
Calculation of coherent modes seen at a wide-band pick-up (BPM)
h
• The patterns of the head-tail modes (m,l) depend on chromaticity
Q’=0
Q’≠0
 m=1 and l=0 
 m=1 and l=1 
 m=1 and l=2 
40
Coherent modes measured at a wide-band pick-up (BPM)
m=1 and l=0
1
Instabilities are a good diagnostics tool to identify and quantify the main
impedance sources in a machine.
41
Wake field (impedances) in accelerator ring design
The full ring is usually modeled with a so called total impedance made
of three main components:
• Resistive wall impedance
•
Several narrow-band resonators at lower frequencies than the pipe
cutoff frequency c/b (b beam pipe radius)
•
One broad band resonator at wr~c/b modeling the rest of the ring
(pipe discontinuities, tapers, other non-resonant structures like
pick-ups, kickers bellows, etc.)
 The total impedance is allocated to the single ring elements by
means of off-line calculation prior to construction/installation
 Total impedance designed such that the nominal intensity is stable
We will talk more about wake fields in the
lecture about linear colliders (tomorrow).
For more details on wake fields, derived
from first physical principles see the
excellent book “Physics of Collective Beam
Instabilities in High Energy Accelerators”, A.
W. Chao (freely available; see course web
pages).
Part II
Overview of multi-particle effects
Adapted from G. Rumolo’s slides from
USPAS Course on collective effects
General definition of multi-particle processes in an accelerator or
storage ring
Class of phenomena in which the evolution of the particle beam cannot be
studied as if the beam was a single particle (as is done in beam optics), but
depends on the combination of external fields and interaction between
particles. Particles can interact between them through
• Self generated fields:
 Direct space charge fields
 Electromagnetic interaction of the beam with the surrounding
environment through the beam‘s own images and the wake fields
(impedances)
 Interaction with the beam‘s own synchrotron radiation
• Long- and short-range Coulomb collisions, associated to intra-beam
scattering and Touschek effect, respectively
• Interaction of electron beams with trapped ions, proton/positron/ion beams
with electron clouds, beam-beam in a collider ring, electron cooling for ions
Multi-particle processes are detrimental for the beam (degradation and
loss, see next slides)
Several names to describe these effects
‚Multi-particle‘ is the most generic attribute. ‚High-current‘, ‚high-intensity‘, ‚high
brightness‘ are also used because these effects are important when the beam has a
high density in phase space (many particles in little volume)
Other labels are also used to refer to different subclasses
• Collective effects (coherent):
 The beam resonantly responds to a self-induced electromagnetic excitation
 Are fast and visible in the beam centroid motion (tune shift, instability)
• Collective effects (incoherent):
 Excitation moves with the beam, spreads the frequencies of particle motion.
 Lead to particle diffusion in phase space and slow emittance growth
• Collisional effects (incoherent):
 Isolated two-particle encounters have a global effect on the beam dynamics
(diffusion and emittance growth, lifetime)
• Two-stream phenomena (coherent or incoherent):
 Two component plasmas needed (beam-beam, pbeam-ecloud, ebeam-ions) and the
beam reacts to an excitation caused by another „beam“
The performance of an accelerator is usually limited by a multi-particle effect. When the beam
current in a machine is pushed above a certain limit (intensity threshold), intolerable losses or beam
quality degradation appear due to these phenomena
Direct space charge forces (as discussed in more details earlier)
x
Transverse space charge
•
Force decays like 1/g2
•
It is always repulsive
y
Longitudinal space charge
dp/p0
z
•
Force decays like 1/g2
•
it can be attractive above transition
Wake fields, impedances (as discussed in more details earlier)
z
W0(z)
L
e
q
s
Model:
A rigid beam with charge q going through a device of length L leaves behind
an oscillating field and a probe charge e at distance z feels a force as a
result. The integral of this force over the device defines the wake field and
its Fourier transform is called the impedance of the device of length L.
Single bunch versus multi-bunch effets
• Single or multi-bunch behavior depends on the range of action of the wake fields
 Single bunch effects are usually caused by short range wake fields (broad-band
impedances)
 Multi bunch or multi-turn effects are usually associated to long range wake
fields (narrow-band impedances)
Wake decays over a bunch length 
Single bunch collective interaction
Wake decays over many bunches 
Coupled bunch collective interaction
W
W(z-z‘)l(z‘)dz‘
W(kd)N(kd)
z
s
d
s
W
Electron cloud
Principle of electron multipacting:
Example of LHC
Electron multiplication is made possible by:
 Electron generation due to photoemission, but also residual gas ionization
 Electron acceleration in the field of the passing bunches
 Secondary emission with efficiency larger than one, when the electrons hit the inner pipe
walls with high enough energy
Example of coherent vs. incoherent effects
Coherent: coherent synchrotron radiation
(CSR)
• Previous calculations of synchrotron
radiation in this course assumed each
particle radiates independently
 Prad  N
•
•
•
Incoherent: intra-beam scattering
• Particles within a bunch can collide with
each other as they perform betatron and
synchrotron oscillations. The collisions lead
to a redistribution of the momenta within
the bunch, and hence to a change in the
emittances.
If particles are close with respect to the
radiation wavelengths, the particles will
radiate coherently (as one macro particle),
 Prad  N2
Significant effect for short bunch lengths,
low energy beams, with large number of
particles per bunch
May lead to instabilities
•
This is effect is called IBS, intra-beam
scattering. If there is a large transfer of
momentum into the longidunal plane. It
is called Touchek scattering.
•
May lead to emittance growth and
reduced beam life time.
More examples
Bunch in the CERN SPS synchrotron
BNL-RHIC, Au-Au operation, Run-4 (2004)
16h
Coherent effects:
When the bunch current exceeds a certain
limit (current threshold), the centroid of
the beam, e.g. as seen by a BPM, exhibits
an exponential growth (instability) and the
beam is lost within few milliseconds
Multi-particle
effects
Collective
Transverse
Longitudinal
Ecloud/trapped
ions
Multi- bunch
Microwave/tur
bulent
Head-tail
Incoherent/
collisional
Two-stream
TMCI
Instability/beam loss
Potential well
distortion
Coherent tune shift
Beam/beam
IBS/Touschek
Incoherent tune spread
Beam quality degradation/
emittance growth